Saturday, May 31, 2014

The Dutch teleportation advance

Most of the generic science news sources – see e.g. PC Magazine – report on a new result by experimenters at a Kavli-named institute in Delft, a historical academic town in Holland, that just appeared in Science:

They have made some progress in the experimental work that could be useful for quantum computers in the future – potentially but not certainly foreseeable future. Two qubits – electron spins somewhere in two pieces of a diamond that are 10 feet away – are entangled, stored as nuclear spins, guaranteed to be sufficiently long-lived, and measured to be almost perfectly entangled.

I don't follow every experimental work of this kind and I won't pretend that I do. This prevents me from safely knowing how new their work is. I hope and want to believe it is sufficiently new, indeed. Obviously, quantum computers will require us to master many more operations than this one – potentially and probably more difficult ones.

However, the words chosen in the paper and in the popularization of the result are a mixed bag.

There have been too many recent TRF blog posts about quantum foundations so I will be brief. The popular articles usually correctly point out that experiments of this sort clearly show that Einstein was wrong etc.

On the other hand, they also tend to present "teleportation" as an actual transfer of information. Even the abstract of the Science paper starts with a comment about a "robust quantum information transfer". There is no actual transfer of information here in entanglement, however. Indeed, such long-distance influences would require unbelievably strong forces and their faster-than-light speed would contradict the laws and symmetries of the special theory of relativity.

What do the entangled bits look like? They are described by the state vector of the form\[

\] where the two characters inside the ket-vectors describe the state of the electrons in the left diamond and the right diamond, respectively. You see that if the left diamond spin is "up" (counter-clockwise if you look from the bird perspective), the right diamond spin is guaranteed to be "down", and vice versa.

Such a perfect anticorrelation could be achieved in classical physics, too. But the spins would have to objectively be either in the \((\uparrow\downarrow)\) classical state or the \((\downarrow\uparrow)\) state before the measurements of the spins. However, if this were so, one would easily see that if we perform measurements of the spin with respect to another pair of axes, like a horizontal axis, the outcomes \((\leftarrow\leftarrow)\), \((\leftarrow\rightarrow)\), \((\rightarrow\leftarrow)\), \((\rightarrow\rightarrow)\) would have the probability 25 percent each. It's because the two spins have to be independent.

You may have noticed that I didn't use the ket vector notation but rather parentheses in the previous paragraph. It is no mistake. On the contrary; it was absolutely deliberate because the states represented configurations in classical physics, not quantum states, and I used the logic of classical physics – its basic framework – to deduce what would be observed. Too many deluded people are using the word "quantum" and sometimes even quantum symbols but they still think classical. They still think that the physical system must be described by an objectively correct classical information at every moment, even before the measurements. That's too bad. If you aren't answering all questions in science – in principle – according to the quantum rules that compute probabilities from complex amplitudes obtained from bra and ket vectors and actions by linear operators and that urge you to avoid any "visualizations" of the "actual arrangement of the system" before the measurement, then you are not doing quantum mechanics, you are not thinking in agreement with modern physics, and you shouldn't use any words that quantum mechanics brought us! You should only use the vocabulary and symbols up to the 19th century because this is where you are hopelessly stuck and you only drown yourself in mud if you are using words you can't possibly understand.

Quantum mechanics says something else. Even in this different basis used for the spins, the electrons are perfectly (anti)correlated because by linear combinations of the basis vectors (applied twice, using the distribution law etc.)\[

\] The relative phase, the absolute phase, and even the question whether it's a perfect correlation or perfect anticorrelation depends on some conventions for the basis vectors and the directions of the axes. I don't want to torture you with my particular choice here but I hope that my conventions have been internally consistent and if you rewrite the vertical spin basis in terms of the horizontal one (using the same absolute directions and the same formulae for both spins), you will get the result I wrote down.

But the more general and more important point is that instead of the 25%,25%,25%,25% distribution predicted by classical physics, one gets a perfect (anti)correlation i.e. 0%,50%,50%,0% again!

This (anti)correlation is guaranteed to be observed, so the reduced \(\ket{\psi_B}\) for the other spin may be immediately deduced from the measurement of the electron at \(A\). As long as you agree that relativity holds and material objects or genuine information can't propagate faster than light, the fact that \(\ket{\psi_B}\) instantaneously changed implies that \(\ket\psi\) encodes subjective information only. The change has only occurred in your head. You learned something about the spin \(A\), and because you knew that the spin \(B\) had a perfectly anticorrelated state, it follows that you may say what the spin at \(B\) is going to be.

Conceptually, this "collapse" is nothing else than the transition from the original complicated multivariate probability distributions such as \(p_{s_1,s_2}\) here or \(\rho(x_A,x_B)\) elsewhere (that exist even in classical statistical physics) to conditional probabilities such as \(p_{\leftarrow,s_2}\) here (or probability distributions of the kind \(\rho(17,x_B)\) elsewhere) that take the already known results of measurements ("conditions") into account. This transition to simplified probability distributions also occurs in your head only. It is not just an analogy. All the actual probability distributions that may be extracted from the state vector \(\ket\psi\) or \(\rho\) in quantum mechanics are doing exactly what the usual "adjustment of probabilities" is expected to do in Bayesian inference. Bayesian inference isn't dependent on classical physics in any way. It's a way of logical, probabilistic thinking and it's fully operational and unmodified in quantum mechanics, too.

This perfect (anti)correlation follows from the structure of the state \(\ket\psi\). And the reason why this state \(\ket\psi\) has the form it has isn't hiding in any event right before the measurements. Instead, the state \(\ket\psi\) has this form because that's how the two electron spins in the diamond were prepared. In the past, the spins came from the same small region of space – they were in a mutual contact. For example, \(\ket\psi\) is the only "singlet" i.e. two-spin state that has the total angular momentum \(|\vec J|^2=0\). So if you guarantee that the two spins are created from an object without any angular momentum, you automatically guarantee the perfect entanglement – the perfect (anti)correlation regardless of the type of the measurement that is made (if you make the same measurement on both spins).

So the reason behind all these correlations – correlations implied by quantum entanglement – is the same as the reason behind the anticorrelation of the two Bertlmann's socks. The color of the socks was decided by Herr Bertlmann in the morning, at the same piece of his brain, so that it was guaranteed that the two socks didn't have the same color (because he is a whackadoodle). The "only" new feature of the entanglement is that the observables in quantum mechanics don't commute with each other which may, perhaps ironically but demonstrably, increase (but also decrease) the degree of (anti)correlation between various pairs of quantities of the two electrons. Classical physics – a framework of physics that assumes the world to be perfectly described by some "objective reality" even in between any two measurements – wouldn't allow that but it's perfectly OK because classical physics is wrong.

Accuracy of the states

Now I want to mention something about the accuracy of the state \(\ket\psi\), the reliability of the correlation. Errors may arise

because the initial state isn't prepared exactly in the idealized state I described or because the state deteriorates between the production or the measurement or

because the two individual electron spins aren't measured exactly

Obviously, the accuracy can't be really 100.0000000000000%. But it may be much better than you expect. What do the errors do?

The Hilbert space of the two qubits is \(2\times 2\) i.e. four-dimensional. The entangled state \(\ket\psi\) I started with may be viewed as one basis vector of this four-dimensional space. It's likely that if you prepare the two spins in this state, there will be a small admixture of the other states, including those that are not anticorrelated. This admixture may also be created by some evolution between the production and the measurement – some deterioration of the state, so to say. The admixture will have coefficients of order \(\alpha,\beta\ll 1\):\[

\] You may think of \(\alpha,\beta\) as some small angles (in radians) with added phases in the four-dimensional complex Hilbert space. Imagine that \(\alpha,\beta\sim 10^{-10}\) because angles, at least if they're actual geometric angles, may be prepared this accurately.

The point I want to make is that if you measure the spins in the realistic initial state \(\ket{\psi'}\), you will sometimes – very rarely – see that the two spins are not anticorrelated because of the two new contributions where the spins are aligned rather than opposite.

But the key point is that \(\alpha,\beta\) are probability amplitudes so the actual probabilities that you will find both spins up or both spins down will be \[

\] and these probabilities are of order \(10^{-20}\), much smaller than the initial angles! Exactly the same comments apply to the imprecision in the final measurements of the spins. Imagine that you measure the spin of the electrons with respect to the axis \(z'\) that isn't exactly vertical but instead, is tilted at a small angle \(\alpha\) or \(\beta\) relatively to the vertical axis \(z\). It is equivalent to adding terms (ket vectors with the "wrong spin") of order \(\alpha\) or \(\beta\) to the state vector, too.

Again, the probability that you measure the "wrong" spin scales like the second power of the small angle!

So the probability of a mistake is really, really small.

I think that this point about quantum mechanics isn't being emphasized sufficiently often but I have started to do so. Quantum mechanics is often presented as being "more fuzzy" than classical physics, in all respects. But it ain't really true. Quantum mechanics is a different theory than classical physics. In some respects, it may be more fuzzy but in others, it is much more accurate.

If you take an object in classical physics and rotate it by a tiny angle \(\alpha\) around some axis, the errors in all generic observables will be of order \(\alpha\), right? The exactly vertical vector should have \(x,y\)-components equal to zero but the rotation makes them of order \(\alpha\). They enter linearly into many things that matter (although others may depend on \(x^2\) only if the linear terms happen to cancel) so you will find effects that were vanishing in the exactly vertical case but that will be non-vanishing, of order \(\alpha\), in the slightly tilted case.

However, if you consider a quantum system with a discrete spectrum, e.g. the spin, the rotation of a system by a small angle \(\alpha\) only changes some vanishing probability amplitudes from zero to \(O(\alpha)\) and it means that the actual probability of all results of a measurement or processes will be of order \(O(\alpha^2)\), much smaller than in classical physics! Quantum mechanics will entirely prohibit any errors whose probability would already scale as \(\alpha\).

So because of this squaring, quantum mechanics actually makes the behavior of physical systems with a discrete spectrum much more immune towards small transformations and rotations that may arise as errors. This is an example of the meme that quantum mechanics may be much less fuzzy than classical physics would ever allow.

The exact values of the discrete eigenvalues – e.g. the energy of an atom – are also something that classical physics could never achieve. Every fluctuation would always be allowed in classical physics. That's why nothing like atomic clocks that can measure time with the accuracy of \(10^{-16}\) would probably ever be built in a classical world!

You may also see that quantum mechanics is able to be "equally as" or "more determined than" classical physics on some quantum games. One may prepare initial states that guarantee some perfect correlations or anticorrelations that directly contradict with what you could derive from the classical logic. The wisdom of these games – infinitely many games – captures all the surprises in Bell's theorem but the contradiction becomes much cleaner and more strict.

(In general, people who are obsessed with Bell's theorem are lousy thinkers for whom group think and parroting of other lousy thinkers is much more likely than a penetrating individual thinking. It's just one particular setup where quantum mechanics shows predictions not reproducible by a classical model. But the truth is that virtually every small quantum system is predicted to behave differently by quantum mechanics than it is by classical physics. And the Bell's setup is neither among the cleverest ones nor among the sharpest or most impressive ones.)

Quantum mechanics is very different and despite its undeserved "fuzzy" image, its being different is actually making many things – including CHSH games, atomic clocks, and entangled pairs – more robust, sharper, and more immune against errors and drifting. Many things that are as sharp as they are in the world around us – perhaps including the reliable RNA/DNA code – would arguably never reach this level of accuracy in a world governed by a classical theory. I mean any classical theory – a theory based on the continuous evolution of objective variables as functions of time.

Thanks, quantum mechanics, for the life and everything that just works in our world.

I wonder if there's proof yet that quantum computing isn't already being used by biology. Photosynthesis for instance is extremely fine tuned at a similar level of particle and photon interaction. I hesitate to mention Penrose's microtubule quantum computer idea, here though. A search for "microtubule" here has no results yet anyway.

I wonder if there's proof yet that quantum computing isn't already being used by biology. Photosynthesis for instance is extremely fine tuned at a similar level of particle and photon interaction. I hesitate to mention Penrose's microtubule quantum computer idea, here though. A search for "microtubule" here has no results yet anyway.

Hi Lubos:This was a nice and rigorous blog on entanglements. While we are on the subject of quantum computers, I have a question. Have you heard about a Canadian company called D-wave which has expanded into California? There was a long article in Time magazine in Feb. 17 issue that they have produced quantum computers with 512 cubits and they have sold 5 machines to defense contractors, NSA and google! But they are keeping it as a trade secret! All of us would agree that one should not try to learn physics from Time magazine! But I am curious. It seems most unlikely that they have entangled 512 cubits, since last I heard was that the present limit is about 8-12 cubits. But they claim that it is faster than classical computers. It may be a quantum simulator and not a full fledged quantum computer. Any idea what is going on?

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Unless some new breakthrough has occurred, the D-Wave gadget isn't a universal quantum computer that could run the Shor algorithm, for example. Instead, it is a case of quantum annealing, something that uses quantum mechanics to improve the search for maxima of functions. So it's something between very useful and very useless.

Well, it may depend on the depth one expects. But I would say that even the key geniuses like Newton, while their advances were amazing and while they were arguably the most intelligent physicists in the history, were thinking superficially - the opposite of deeply - about reality because they were assuming that they could directly read the laws of physics from what they see in the everyday life.

This is not what I call a deep analysis today - it's a superficial analysis - and as I said, I am also convinced that such naive laws based on the classical-physics-based everyday experience wouldn't be able to run a world where things may change continuously yet protect discrete information with a sufficient reliability that is needed e.g. for life.

No analysis today would make any sense without quantum mechanics, but surely no one is suggesting we should start pretending that quantum mechanics hasn't been discovered.

The issue is that of prooper perspective on the depth of insight that went into the advances of the 17th-19th centuries, plus the special and general relativity, which completed the classical era in the early 20th century.

Not all of classical physics was mere phenomenology. Theory of elasticity, hydrodynamics, and theory of heat conduction are examples of classical phenomenological theories that provide no insight into the nature of reality, but which in any case have attracted mathematicians more than physicists even in their early days (Cauchy, Fourier, Euler, etc.).

On the other hand, I don't see how Maxwell's theory, built on Faraday's concept of a field, can be regarded as phenomenological in its outlook at the time it was created. The same goes for special and general relativity.

Figuring out that light was an electromagnetic wave was also a significant insight into the nature of reality, not just some "phenomenology".

One can argue whether thermodynamics and statistical mechanics are phenomenological or fundamental. Personally, I view them as too deep and too general to be considered phenomenological the way elasticity theory or hydrodynamics are phenomenological, but if someone feels otherwise, fine.

There was also plenty of work in the 18th and 19th century that firmed up the atomic picture. A lot of this work was done by chemists, but Clausius, Maxwell, and Boltzmann, among others, employed it in physics.

It would be utter folly, Swine flu, to argue that thermodynamics and statistical mechanics are not fundamental. There have been geniuses scattered throughout the history of physics and they clearly did think deeply but there were, even in 1925, many questions that were not answerable, not even in principle. Now we can see the path to a theory of everything and that vision is all about quantum mechanics, statistical mechanics (and gravity).

The pictures are very nice. Some techniques probe the electron density, others probe things closer to the molecular electrostatic potential. You can also infer the shape of Dyson orbitals (difference between the N-electron wave function and the N-1 electron wave function) via scattering of an ion with its own previously ionized electron. See here: “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867–871 (2004).

No one doubts that through many measurements of noncommuting operators you can "reconstruct" the density matrix, or more likely the 1- or 2-reduced density matrices. Wave functions have real consequences after all. But if you want maximal information, it is not enough to "measure" (or rather probe) x or p, that is why you cannot only look (observe) one single thing in your experiment. In any case you have to gather many different pieces of information and then through a computer (or your brain) reconstruct (learn) how the wave function is.

There is no objective thing in the sense of classical physics that would cause electron's interference, or that would cause anything else in the Universe, for that matter, so all attempts to classically visualize what's going on are ultimately wrong. Hasn't this question been answered about 6,000 times already?

Only the crackpots can draw a crowd of common people. The 99% of 'normal' physicists won't get any attention: (The physicists who understand quantum physics aren't interesting to common people). (No point in having a discussion forum if all of the participants agree... 'So what's wrong with Quantum Physics?' ... 'Nothing.' 'Nothing.' 'Nothing.' 'Nothing.' .... 'Really, no one here disagrees with someone else who is here?' ... 'No.' 'Not really.' 'No, we don't disagree.') (Conflict that doesn't exist doesn't draw an audience.)

Now maybe the answer is "we can't possibly know." That would be fine. In my field we have things like "busy beaver" machines where we can describe what they calculate, but we can't possibly do the calculation. Likewise there's Chaiten's Ω, which is also describable but not computable. So, I'd be perfectly happy if the behavior of an electron is describable, but what an electron is cannot possibly be imagined (because our brains are Turing machines and we can't compute what an electron is. Don't know how to prove this, though).

If this is the case, it would help if physicists would only talk about electrons based on their behavior and not their ontology.

Dear Bob, I translated your "physical object" to a more accurate "classical physics object" because it is absolutely self-evident that this is what you meant.

If you talk about physical objects without any association with classical physics, it is spectacularly clear what objects exist there - where "XY exists" is meant to be a proposition whose truth value is interpreted according to the laws of physics.

The objects that exist in this experiment are the electron, and the barrier with two slits. It's as simple as that. There is obviously no other physical object involved. But the behavior of the electron in the environment with the two slits is obeying the laws of physics and the laws of physics imply that the electron will land at a random place so that many electrons will fill an interference pattern.

One doesn't *need* any additional "physical objects" and additional "physical objects" wouldn't help to do what is being instead being done by the laws of physics.

Lubos, you are mentioned in Spain. For a 2013 post where you called "spanish-crank" to a physicist who now is leftist deputy in the last european elections. Quantum on-topic. http://francis.naukas.com/2014/06/01/el-fisico-y-eurodiputado-pablo-echenique-robba-es-un-negacionista-de-la-cuantica/

Lubos, you are mentioned in Spain. For a old post where you called quantum-crank to a someone who now is leftist deputy in the last european elections. http://francis.naukas.com/2014/06/01/el-fisico-y-eurodiputado-pablo-echenique-robba-es-un-negacionista-de-la-cuantica/

Dear Lubos,the quantum qbism is ok and well of course.But for me the consistent histories approach which you promoted is a lot more interesting because it broadens the view from the measurement situation which is a very special situation in nature to the more general question how an approximate classical world can emerge from the laws of quantum mechanics.

The pilot wave have got the support from the walking droplets` fluid mechanics.Some theorists get the quantum behaviour of the particles moving under the attack of some chaotic gase of the vacuum energy fluctuation. The determinism returnes?